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Chapter 3: Problem 1

Without using a calculator or computer, give a rough estimate of \(2^{83}\).

Short Answer

Expert verified
A rough estimate of \(2^{83}\) is \(8 * 10^{24}\) or 8 followed by 24 zeros.

Step by step solution

01

Find a nearest, easily estimable power of 2

A nearby power of 2 whose value is easy to estimate is \(2^{80}\). This is because the number 80 is divisible by 10, and we can use the property \(2^{10} = 1024 \approx 1000\).
02

Use exponent properties to break down \(2^{83}\)

Now we can rewrite the given expression \(2^{83}\) using exponent properties: \(2^{83} = 2^{80} * 2^3\)
03

Estimate \(2^{80}\)

We know that \(2^{80}\) is equivalent to \((2^{10})^8\). Because we can consider \(2^{10} \approx 1000\), we obtain: \((2^{10})^8 \approx (1000)^8\)
04

Estimate \(2^3\)

The value of \(2^3\) is easy to calculate without a calculator: \(2^3 = 8\)
05

Combine the estimates to find an approximation for \(2^{83}\)

Now we can use the estimates for \(2^{80}\) and \(2^3\): \(2^{83} \approx (1000)^8 * 8\)
06

Simplify the approximation

Finally, we can simplify the approximation: \(2^{83} \approx 8 * (1000)^8 = 8,000,000,000,000,000,000,000,000\) So, a rough estimate of \(2^{83}\) is \(8 * 10^{24}\) or \(8\) followed by 24 zeros.

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Show that the range of cosh is the interval \([1, \infty)\).

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